Math 112 Test 2 Non-calculator part Practice Problems - Summer 2012
Divide using long division.
8x3 + 18x2 + 13x + 5
1)
-4x - 3
2) (5x5 - x3 - 4x2 - 233x + 28) ÷ (x2 - 7)
Divide using synthetic division.
3) (x2 + 8x + 9) ÷ (x + 5)
4)
x5 + x3 - 5
x - 2
Use synthetic division and the Remainder Theorem to find the indicated function value.
5) f(x) = x4 - 7x3 - 6x2 - 7x + 3; f(-3)
6) f(x) = 7x4 + 3x3 + 6x2 - 4x + 73; f(2)
Find a rational zero of the polynomial function and use it to find all the zeros of the function.
7) f(x) = x3 + 2x2 - 9x - 18
8) f(x) = 3x3 - 14x2 + 13x + 6
Find an nth degree polynomial function with real coefficients satisfying the given conditions.
9) n = 3; 3 and i are zeros; f(2) = 30
10) n = 3; - 4 and i are zeros; f(-3) = 60
Use the graph of the rational function shown to complete the statement.
11)
10
y
8
6
4
2
-10 -8 -6 -4 -2
-2
2 4
6 8 10
x
-4
-6
-8
-10
As x→2 - , f(x)→ ?
As x→2 + , f(x)→?
1
Find the vertical asymptotes, if any, of the graph of the rational function.
x
12) g(x) = x + 2
13) h(x) = x + 2
x(x + 5)
14) h(x) = x + 3
x2 + 9
15)
x - 25
x2 - 8x + 15
16) h(x) = x + 3
x2 - 9
Find the horizontal asymptote, if any, of the graph of the rational function.
15x
17) f(x) = 3x2 + 1
18) g(x) = 6x2
2x2 + 1
19) h(x) = 25x3
5x2 + 1
20) g(x) = 7x2 - 6x - 4
2x2 - 5x + 2
1
1
Use transformations of f(x) = or f(x) = to graph the rational function.
x
x2
21) f(x) = 1
+ 2
x - 2
y
10
5
-10
-5
5
10
x
-5
-10
2
22) f(x) = 1
+ 2
(x - 5)2
y
10
5
-10
-5
5
10
x
-5
-10
Graph the rational function. Be able to give the domain, the y -intercept, the x-intercepts, equations of vertical
asymptotes, and equations of horizontal asymptotes.
5
23) f(x) = 2
x + 6x + 9
y
10
5
-10
-5
5
10
x
-5
-10
24) f(x) = x2
x2 - x - 56
y
x
3
Solve the polynomial inequality and graph the solution set on a number line. Express the solution set in interval
notation.
25) x2 - 5x + 4 > 0
-9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9
26) x2 - 5x - 14 < 0
-9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9
Solve the rational inequality and graph the solution set on a real number line. Express the solution set in interval
notation.
x - 2
27)
< 0
x + 1
-9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9
28)
-x + 9
≥ 0
x - 5
-10-9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10
Find the inverse of the one-to-one function. You will need to show each of the four steps.
3
29) f(x) = 8x - 5
Find the inverse of the one-to-one function.
30) f(x) = x + 4
31) f(x) = (x + 2)3
Does the graph represent a function that has an inverse function?
32)
y
x
4
33)
y
x
Use the graph of f to draw the graph of its inverse function.
34)
y
10
5
-10
-5
5
10
x
5
10
x
-5
-10
35)
y
10
5
-10
-5
-5
-10
5
Graph the function by making a table of coordinates.
36) f(x) = 5 x
6
y
4
2
-6
-4
-2
2
4
6 x
2
4
6 x
-2
-4
-6
37) f(x) = 1 x
2
6
y
4
2
-6
-4
-2
-2
-4
-6
Write the equation in its equivalent exponential form.
38) log 64 = 3
4
39) log x = 3
5
Write the equation in its equivalent logarithmic form.
40) 6 3 = x
41) b4 = 2401
Evaluate the expression without using a calculator.
42) log 4
2
43) log
7
7
44) log3 243
6
Evaluate or simplify the expression without using a calculator.
45) log 10,000
46) ln e
47) ln 1
48) ln e3
Evaluate the expression without using a calculator.
1
49) log6 6
50) log 1
5
51) log 9
9
Find the domain of the logarithmic function.
52) f(x) = log (x + 3)
5
53) f(x) = ln (4 - x)
54) f(x) = log (x + 7)2
8
55) f(x) = log (x2 - 5x + 6)
Evaluate the expression without using a calculator.
56) eln 235
57) ln e8x
7
Answer Key
Testname: UNTITLED1
1) -2x2 - 3x - 1 + 2) 5x3 + 34x - 4 + 3) x + 3 - 2
-4x - 3
22)
y
10
5x
x2 - 7
5
6
x + 5
35
4) x4 + 2x3 + 5x2 + 10x + 20 + x - 2
-10
-5
5
10
x
5
10
x
-5
5) 240
6) 225
7) {-3, -2, 3}
1
8) - , 2, 3
3
-10
23)
9) f(x) = -6x3 + 18x2 - 6x + 18
10) f(x) = 6x3 + 24x2 + 6x + 24
y
10
11) f(x)→+∞
f(x)→-∞
12) x = -2
13) x = 0 and x = -5
14) x = 3
15) x = 5, x = 3
16) x = 3
17) y = 0
18) y = 3
19) no horizontal asymptote
7
20) y = 2
5
-10
-5
-5
-10
24)
y
6
21)
5
y
4
10
3
2
1
5
-16
-10
-5
5
10
-8
-1
8
16
x
-2
x
-3
-4
-5
-5
-6
-10
25) (-∞, 1) ∪ (4, ∞)
-9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9
26) (-2, 7)
-9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9
8
Answer Key
Testname: UNTITLED1
27) (-1, 2)
36)
6
y
-9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9
4
28) (5, 9]
2
-10-9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10
3
5
29) f-1 (x) = + 8x 8
-6
-4
-2
2
4
6 x
2
4
6 x
-2
30) f-1 (x) = x2 - 4
3
31) f-1 (x) = x - 2
32) No
33) Yes
34)
-4
-6
37)
6
y
y
4
10
2
5
-6
-10
-5
5
10
-4
-2
-2
x
-4
-5
-6
-10
38) 4 3 = 64
39) 5 3 = x
35)
40) log
x = 3
6
41) logb 2401 = 4
y
10
42) 2
1
43)
2
5
-10
-5
5
10
44) 5
45) 4
46) 1
47) 0
48) 3
49) -1
50) 0
51) 1
52) (-3, ∞)
53) (-∞, 4)
54) (-∞, -7) or (-7, ∞)
55) (-∞, 2) ∪ (3, ∞)
56) 235
57) 8x
x
-5
-10
9